An Incident Related To The Life Of The Prophet (sws)

In April I became a Muslim. I've been attempting to learn as much as I can about this faith. Before I converted, I studied for a few months, but it is impossible to read everything. Recently I've run across something that makes me doubt the truth of Islam, and I would like it cleared up. It is hurting my iman. I have gone through so much to accept this faith, and I don't want to let it go. Please, explain the meaning of this hadith to me. It seems brutal and contrary to everything I believe about Allah and his prophet (sws).

Narrated Abdullah Ibn Abbas:
A blind man had a slave-mother who used to abuse the Prophet (sws) and disparage him. He forbade her but she did not stop. He rebuked her but she did not give up her habit. One night she began to slander the Prophet (sws) and abuse him. So he took a dagger, placed it on her belly, pressed it, and killed her. A child who came between her legs was smeared with the blood that was there. When the morning came, the Prophet (sws) was informed about it. He assembled the people and said: I adjure by Allah the man who has done this action and I adjure him by my right to him that he should stand up. Jumping over the necks of the people and trembling the man stood up. He sat before the Prophet (sws) and said:
Apostle of Allah! I am her master; she used to abuse you and disparage you. I forbade her, but she did not stop, and I rebuked her, but she did not abandon her habit. I have two sons like pearls from her, and she was my companion. Last night she began to abuse and disparage you. So I took a dagger, put it on her belly and pressed it till I killed her. Thereupon the Prophet (sws) said: Oh be witness, no retaliation is payable for her blood.
Here are links to the Hadith:
The USC website is used frequently to quote ahadith on the web. The other, www.jannah.com is also a respected Islamic website. Is this hadith authentic? How do I find out?

Thanks so much for any light you can shed on this subject. Is there any source I can find (other than my own opinion) that can discredit this hadith? If this hadith isn't valid, why is it among the collections of hadith? Why hasn't it been discarded?

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